Question

Whats the most efficient Pokeball for catching Mew?

Comments

Go with the Heavy Ball since Mew's the heaviest Pokemon in existence.

Answer 1

The most efficient Pokeball for catching any Pokémon is the Master Ball. It has a 100% catchrate and will never fail.

Outside of that, Dusk Balls (at night) and Ultra Balls are decent alternatives, but they are nowhere near as efficient as the Master Ball.

Comments

Dusk balls don't exist in any of the same games as wild Mews, and ultra balls are worse than net balls.

Answer 2

This assumes you don't want to use a master ball, and you're trying to catch the Faraway Island Mew, because that's the only place with wild Mews.
source
If both assumptions are true, then the best ball would be the net ball. The key here is that, when Mew transforms, it copies the target's types and catch rate. The easiest way to catch Mew is to use a water or bug Pokemon with 255 catch rate (guaranteed catch with net balls), or one of these: Caterpie, Weedle, Poliwag, Magikarp, Ledyba, Spinarak, Wooper, Wurmple, Lotad, Nincada, Feebas, Spheal, Clamperl Use it to battle the Mew, and throw a net ball after the Mew transforms, so you'll get a guaranteed catch. If the Pokemon faints before the Mew transforms, simply soft-reset.